Approaches over a foreign country

How many airports are there where you temporarily cross the
border during the approach?

I notice that Tijuana has an approach path over the SW
tip of the U.S., while Brownsville has an approach
over part of Mexico. El Paso also appears to have
approaches over Ciudad Juarez, but I have no idea
whether these are flown as short finals (remaining
in the U.S.)

Does each approach involve the foreign ATC agency, or
are there letters of agreement where, say, Brownsville
TX approach can control part of Mexico's airspace?

Mark